# Definition:Möbius Strip

## Definition

A Möbius strip is a surface with boundary obtained by twisting a side of a rectangle by $180$ degrees, then joining the twisted side and the opposite side of it together.

Thus in the above diagram, $AB$ is identified with $CD$.

### Formal Construction

Let $T$ be the square embedded in the Cartesian plane defined as:

$T = \closedint 0 1 \times \closedint 0 1$

Let $T'$ be the quotient space formed from $T$ using the identification mapping $p: T \to T'$ as follows:

$\forall \tuple {x, y} \in T: \map p {x, y} = \begin {cases} \tuple {1, 1 - y} & : x = 0 \\ \paren {x, y} & : x \ne 0 \end {cases}$

Then $T'$ is a Möbius strip.

## Also known as

Some sources use the term Möbius band, presumably on the grounds that the word strip may be considered inappropriate language to use in front of children and adolescents.

In typographic environments where it is impossible or inconvenient to implement the umlaut, the spelling Moebius strip is often seen.

## Also see

• Results about the Möbius strip can be found here.

## Source of Name

This entry was named for August Ferdinand Möbius.